On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let R be the region formed by the union of the square and all the triangles, and let S be the smallest convex polygon that contains R. What is the area of the region that is inside S but outside R ?
Answer Choices
A. \dfrac{1}{4}
B. \dfrac{\sqrt{2}}{4}
C. 1
D. \sqrt{3}
E. 2 \sqrt{3}